Eigenvalues of diagonal matrix

Question

Problem:

Let $A$ ∈ $\Bbb C^{n×n}$ and let $A$ be a diagonal Matrix with entries $\lambda_1, \ldots, \lambda_n$ ∈ $\mathbb{C}$ Determine spec($A$*$A$)

I think it is clear that the spec ($A$ *$A$) = {$\lambda_1, \ldots, \lambda_n$}, as the eigenvalues of a diagonal matix are just the elements on the diagonal. Could someone appove my thoughts?

Answer 1

In fact when you ll multiply twice the matrix, if you apply an eigenvector, you ll get the eigen value squared. It s quite obvious when you do the computation. So you ll get

spec ($A$ *$A$) = {$\lambda_1^2 , \ldots, \lambda_n^2 $}

Moreover, I think this is true for any matrix, not only diagonals one.

Answer 2

$A^*A= diag(|\lambda_1|^2,....,|\lambda_n|^2)$

Thus the eigenvalues of $A^*A$ are ?